You almost have it: the participle is being used substantively here, with its own object: τον ταδ' ἠγγελκοτα / the one who announced these things.It is the object of τιμᾱ (from τιμάω)ξενιᾴ τε δωροισ τε are instrumtal datives

The first, because of καιπερ = "although". You don't need the [], though, since σε is there.

Also, to get the second meaning, I believe you would need the article with the participle. Generally speaking when the participle is equivalent to a relative clause and is definite it's used with the article.

So why is a participle being used, when it seems to be functioning exactly like a verb? It doesn't seem to have an adjectival quality in this instance. Why not just kaleo in the first person singular active indicative?

In general, Greek prefers to use participles where English would use a subordinate clause with a finite verb, and it's just something you have to get used to. It's similar to in English how you might say "Having finished reading, I decided to go to bed" but Greek does it to a much greater extent. It's still adjectival in that it describes the circumstances (it's often called a circumstantial participle) or describes the noun indirectly through the verb. Adjectives, though, do the same thing

ο πρωτος Αθηναιος ειπεν... = the first Athenian (i.e. the Athenian who was first) said...ο Αθηναιος πρωτος ειπεν... = the Athenian was the first to (i.e. the Athenian, being first) said....

Specifically in this case, though, it's just that the most common way of saying "although" is καιπερ + participle.

ον is the ending for the aorist imperative (2nd person singular) -- at least for those aorists that end in -α rather than -ον. I mean, e.g., ηρωτησα > ερωτησον but ελαβον > λαβε.

About the augment, that only occurs in the past tenses of the indicative mood. It doesn't occur in any of the other moods, so you have to always look at the stem. The aorist stem is the same as the future stem a lot of the time and so there's the potential for ambiguity, e.g. ερωτησω is both future indicative and aorist subjunctive, but in context it shouldn't cause too much trouble.

You got the meaning right, so I'll just help with the analysis. πεπεμμένοι here is not the subject but is the complement of the implied subject that goes to back to ξένους. The reason for εἰσίν is that in the passive of the perfect tense, instead of there being a single verb, you use "to be" plus a participle. For example, if the question had been "where they are going", it could be something like

... ὅποι πορεύονται

so πεπεμμένοι εἰσίν as a group functions like πορεύονται.

About ξένους being accusative, the key thing is that for the most part, the case of a noun is determined by its role in the clause it's in. So ξένους is in the main clause and is the direct object of ἐρώτησον so it's accusative, but πεπεμμένοι is in the ὁπόθεν-clause, and there is modified the (implied) subject, so it has to be nominative.

And about τοὺς οὐκ ὄντας Ἕλληνας, here because of the article, it should be "the strangers who are not Greeks". It would be"not being Greeks" if there were no article.